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To localize sound sources in room acoustic environments, time differences of arrival (TDOA) between two or more microphone signals must be determined. This problem is often referred to as time delay estimation (TDE). The multichannel cross-correlation-coefficient (MCCC) algorithm, which is an extension of the traditional cross-correlation method from two- to multiple-channel cases, exploits spatial information among multiple microphones to improve the robustness of TDE. In this paper, we propose a multichannel spatio-temporal prediction (MCSTP) algorithm, which can be viewed as a generalization of the MCCC principle from using only spatial information to using both spatial and temporal information. A recursive version of this new algorithm is then developed, which can achieve similar performance as MCSTP, but is computationally more efficient. Experimental results in reverberant and noisy environments demonstrate the advantages of this new method for TDE.